(slides 3) visual computing: geometry, graphics, and vision

© 2011 Frank Nielsen

INF555

Fundamentals of 3DLecture 3:

Debriefing: Lecture 2Rigid transformations/QuaternionsIterative Closest Point (+Kd-trees) Frank Nielsen

[email protected]

28 Septembre 2011


• Depth discontinuity• Surface orientation

discontinuity• Reflectance

discontinuity (i.e., change in surface material properties)

• Illumination discontinuity (e.g., shadow)

Harris-Stephens' combined corner/edge detector


Harris-Stephens edge detector

Aim at finding good feature


Harris-Stephens edge detectorMeasure the corner response as

Algorithm:– Find points with large corner response function R (R > threshold)– Take the points of local maxima of R


Edge thresholding hysterisis

● If a pixel value is above the high threshold, it is an edge.

● If a pixel value is below the low threshold, it is not an edge.

● If a pixel value is between the low and high thresholds,it is an edge if it is connected to another edge pixel,otherwise it is interpreted as noise.

Two thresholds: low and high

Single threshold value for edges -> Streaking


Edge hysteresis


Homogeneous coordinates and duality point/line

homogenization

Homogeneous vectorInhomogeneous vector

dehomogenization (also known as Perspective division)


Projective plane

Equivalence class:

is equivalent to

Line coefficients stored in an inhomogeneous vector

Equation of the line:

Point and line have same homogeneous representation:A point can be interpreted as the coefficients of the line

Point and line have same homogeneous representation:A point can be interpreted as the coefficients of the line


Intersection of lines

Cross-product of two vectors:

Intersection point of two lines is obtained from their cross-product:

Line passing through two « points » l1* and l2*: l=p*=l1* x l2*

p= l1 x l2

Duality


Application: Detection of line segment intersection

l1=CrossProduct(p,q);l2=CrossProduct(r,s);// intersection point is the cross-product //of the line coefficients (duality)intersection=CrossProduct(l1,l2);intersection.Normalize(); // to get back Euclidean point


Overview of duality in projective geometry

The determinant of three points represent the volume of their parallepiped.


2D Transformations using homogeneous coordinates


Cartesian coordinate systems in 3D


3D Transformations using homogeneous coordinates

Be careful: Gimbal lock


Euler rotation


Cross-product/outer product

Consider the cross-product as a matrix multiplication:

Outer-product


Arbitrary matrix rotation:Rodrigues' formula

Equivalent to:


Rigid transformations

Concatenation (non-commutative!)


Quaternions for rotations

● Easy to invert scalars● For 2D vectors, invert using complex numbers...● For 3D vectors??? (-> 4D quaternions)● For dD vectors??? (-> 8D octonions)

● Easy to invert scalars● For 2D vectors, invert using complex numbers...● For 3D vectors??? (-> 4D quaternions)● For dD vectors??? (-> 8D octonions)

Provide rotation operator that is invertible

Sir William Rowan Hamilton

Lectures on Quaternions

http://digital.library.cornell.edu/


Quaternions: 1D real+3D imaginary

Real part (1D) Imaginary part (3D, i j k vectors)

Multiplication:

Norm (l2)


Unit quaternions

For a given 3D point p, we compute its rotation Rp as

Rotation theta around an axis u: Quaternion representation:

conjugate


Unit quaternions for rotations


Conversion rotation matrix to quaternion


Spherical linear interpolation (SLERP)

LERP is non-sense for rotation matrices:

SLERP is using quaternion algebra:


Spherical linear interpolation (SLERP)

Useful for computer graphics animation(bone, skinning at articulation)


Bilateral filtering

Edge-preserving smoothing

Videos/demo


Gaussian filtering: Blur everything

Traditional spatial gaussian filtering


Bilateral filtering

New! gaussian on the intensity difference filtering

Bilateral Filtering for Gray and Color Images, Tomasi and Manduchi 1998.... SUSAN feature extractor...

http://www.cs.ucsc.edu/~manduchi/Papers/ICCV98.pdf


Domainfiltering

Range filtering


Iterative Closest Point (ICP)

Align point sets. For example, terrains (DEMs)

Robotics


Align point sets obtained from range scanners

http://www-graphics.stanford.edu/projects/mich/


Solve stone jigsaws...

ICP for solving jigsaws


ICP: Algorithm at a glance

● Start from a not too far transformation

● Match the point of the target to the source● Compute the best transformation from point correspondence● Reiterate until the mismatch error goes below a threshold

In practice, this is a very fast registration method...

A Method for Registration of 3-D Shapes. by: Paul J Besl, Neil D Mckay.IEEE Trans. Pattern Anal. Mach. Intell., Vol. 14, No. 2. (February 1992


ICP: Finding the best rigid transformation

Given point correspondences, find the best rigid transformation.

Source/ModelObservation/Target

Find (R,t) that minimizes the squared euclidean error:


Align the center of mass of sets:


Finding the rotation matrix:

Compute the singular value decomposition

Optimal transformation:


Registration of many point sets to a common atlas

Scoliotic Spine (Atlas of 307 patients)

Many variants of ICP method (truncated, robust, etc.)Many variants of ICP method (truncated, robust, etc.)


In theory, ICP may provably run very slowly for well-constructedwell-constructed point sets...

What computational geometers say

David Arthur; Sergei VassilvitskiiWorst-case and Smoothed Analysis of the ICP Algorithm, with an Application to the k-means MethodFOCS 2006 => O(n/d)^d iterations (exponential)

... but smooth analysis of ICP is polynomial


Computing nearest neighbors in ICP...

● Naive linear-time algorithm● Tree-like algorithm using kd-trees● Tree-like algorithm using metric ball trees●...

Challenging problem in very high-dimensions(common to work up to dimension > 1000 nowadays)

Nearest neighbor of q


Installez Processing.org

→ Rendu des TD1, 2 et 3 pour le Mardi 4 Octobre 17h(Upload work sur la page du cours)

(slides 3) visual computing: geometry, graphics, and vision

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